Euclidean geometry

Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements was the first systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions ( theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system.

The Elements begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Many other consistent formal geometries are now known, the first ones being discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.

Axiomatic approach

Following a precedent set in the Elements, Euclidean geometry has been exposited as an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms):

Any two points can be joined by a straight line.

Any straight line segment can be extended indefinitely in a straight line.

Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

All right angles are congruent.

Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and centre, right angle, congruence, inner and right angles, sum. The following verbs appear: join, extend, draw, intersect. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere.

A proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane:

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that probably cannot be constructed within the theory.

The Elements also include the following five "common notions":

Things that equal the same thing also equal one another.

If equals are added to equals, then the wholes are equal.

If equals are subtracted from equals, then the remainders are equal.

Things that coincide with one another equal one another.

The whole is greater than the part.

Euclid also invoked other properties pertaining to magnitudes. 1 is the only part of the underlying logic that Euclid explicitly articulated. 2 and 3 are "arithmetical" principles; note that the meanings of "add" and "subtract" in this purely geometric context are taken as given. 1 through 4 operationally define equality, which can also be taken as part of the underlying logic or as an equivalence relation requiring, like "coincide," careful prior definition. 5 is a principle of mereology. "Whole," "part," and "remainder" beg for precise definitions.

In the 19th century, it was realized that Euclid's ten axioms and common notions do not suffice to prove all of theorems stated in the Elements. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore needs to be an axiom itself. The very first geometric proof in the Elements, shown in the figure on the right, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. His axioms, however, do not guarantee that the circles actually intersect, because they are consistent with discrete, rather than continuous, space. Starting with Moritz Pasch in 1882, many improved axiom systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski.

To be fair to Euclid, the first formal logic capable of supporting his geometry was that of Frege's 1879 Begriffsschrift, little read until the 1950s. We now see that Euclidean geometry should be embedded in first-order logic with identity, a formal system first set out in Hilbert and Wilhelm Ackermann's 1928 Principles of Theoretical Logic. Formal mereology began only in 1916, with the work of Lesniewski and A. N. Whitehead. Tarski and his students did major work on the foundations of elementary geometry as recently as between 1959 and his 1983 death.

The parallel postulate

To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it.

Many geometers tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs had been published, but all were found to be incorrect. In fact the parallel postulate cannot be proved from the other four: this was shown in the 19th century by the construction of alternative ( non-Euclidean) systems of geometry where the other axioms are still true but the parallel postulate is replaced by a conflicting axiom. One distinguishing aspect of these systems is that the three angles of a triangle do not add to 180°: in hyperbolic geometry the sum of the three angles is always less than 180° and can approach zero, while in elliptic geometry it is greater than 180°. If the parallel postulate is dropped from the list of axioms without replacement, the result is the more general geometry called absolute geometry.

Treatment using analytic geometry

The development of analytic geometry provided an alternative method for formalizing geometry. In this approach, a point is represented by its Cartesian (x,y) coordinates, a line is represented by its equation, and so on. In the 20th century, this fit into David Hilbert's program of reducing all of mathematics to arithmetic, and then proving the consistency of arithmetic using finitistic reasoning. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered to be theorems. The equation

defining the distance between two points P = (p,q) and Q = (r,s) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries.

A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with lines) were photographed during a solar eclipse. The rays of starlight were bent by the Sun's gravity on their way to the earth. This is interpreted as evidence in favour of Einstein's prediction that gravity would cause deviations from Euclidean geometry.

As a description of physical reality

Euclid believed that his axioms were self-evident statements about physical reality. However, Einstein's theory of general relativity shows that the true geometry of spacetime is non-Euclidean. For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting the deviations from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the observation of the slight bending of starlight by the Sun during a solar eclipse in 1919, and non-Euclidean geometry is now, for example, an integral part of the software that runs the GPS system. It is possible to object to the non-Euclidean interpretation of general relativity on the grounds that light rays might be improper physical models of Euclid's lines, or that relativity could be rephrased so as to avoid the geometrical interpretations. However, one of the consequences of Einstein's theory is that there is no possible physical test that can do any better than a beam of light as a model of geometry. Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning.

Logical status

Euclidean geometry is a first-order theory. That is, it allows statements that begin as "for all triangles ...," but it is incapable of forming statements such as "for all sets of triangles ..." Statements of the latter type are deemed to be outside the scope of the theory.

We owe much of our present understanding of the properties of the logical and metamathematical properties of Euclidean geometry to the work of Alfred Tarski and his students, beginning in the 1920s. Tarski used his axiomatic formulation of Euclidean geometry to prove it consistent, and also complete in a certain sense: every proposition of Euclidean geometry can be shown to be either true or false. Gödel's theorem showed the futility of Hilbert's program of proving the consistency of all of mathematics using finitistic reasoning. Tarski's findings do not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply. Although Hilbert thought Euclidean geometry could be put on a firmer foundation by rewriting it in terms of arithmetic, in fact Euclidean geometry is complete and consistent in a way that Godel's theorem tells us arithmetic can never be.

Although complete in the formal sense used in modern logic, there are things that Euclidean geometry cannot accomplish. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible.

Absolute geometry, formed by removing the parallel postulate, is also a consistent theory, as is non-Euclidean geometry, formed by alterations of the parallel postulate. Non-Euclidean geometries are consistent because there are Euclidean models of non-Euclidean geometry. For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.

Classical theorems

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